If $S$ is the surface given by the function $z=y^2-x^2$, if I have the points $A=(1,0,-1)$, $B=(0,1,1)$, $C=(1,1,0)$, how can I use the Gaussian curvature to determine if there is an isometry of $S$ that takes $A$ into $B$? or $A$ into $C$? I know that if the Gaussian Curvature is different in, let's say, $A$ and $C$, then there is no isometry that locally takes $A$ into $C$; but what can I say about the other way around? If the curvatures coincide then there exists an isometry of $S$? Can I try to do this by rotating $S$?
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The points A and B does not lie on S – Feb 13 '13 at 18:17
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Sorry, I inverted the function $z$. Just corrected. – user62182 Feb 13 '13 at 18:19
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In general, having the same curvature at a point is not enough to guarantee there is a local isometry between the points. For example, picture a parabaloid. At it's vertex, it has some curvature $K$. Imagine a sphere of radius $K$. Then the curvature at the vertex of the parabola matches the curvature on every point of the sphere, but there is no local isometry between any point on the parabola and any point on the sphere because every neighborhood on the parabola contains points of different curvatures, while no neighborhood on the sphere does. – Jason DeVito - on hiatus Feb 14 '13 at 02:49
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Yes, but I need isometries of S! – user62182 Feb 14 '13 at 10:56